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How did set theory become a universal language for stating mathematical theories?

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logic: universal affirmative — Modern logic Representations of the universal affirmative, “All A's are B's” in modern logic.

Set theory became a universal language for mathematics by providing a precise, adaptable framework capable of defining complex mathematical concepts from a single source. Initially developed by Georg Cantor in the late 19th century, the theory allowed mathematicians to treat infinite sets as rigorous mathematical objects.1

The transition to a universal language was driven by the realization that mathematical notions—including numbers and functions—could be reduced to the basic concepts of set membership and equality. For instance, integers can be defined using the empty set and its successors.1234

While early "naive" set theory faced challenges like Russell’s paradox, the development of axiomatic set theory, particularly Zermelo-Fraenkel set theory, resolved these contradictions by constraining how sets are constructed. This axiomatic approach allowed mathematicians to dispense with informal definitions and cast all of mathematics into a unified structure. As noted in the Nicolas Bourbaki Éléments de mathématique, it is logically possible to derive practically the whole of known mathematics from the theory of sets.1324